**What is Boolean Algebra?**

In 1854, **George Boole, **a 19^{th} century English Mathematician has invented Boolean/ Logical/ Binary algebra by which reasoning can be expressed mathematically.

**Boolean algebra** is one of the branches of algebra which performs operations using variables that can take the values of binary numbers i.e., **0 (OFF/False) **or** 1 (ON/True)** to analyze, simplify and represent the logical levels of the digital/ logical circuits.

**0<1**, i.e., the logical symbol **1** is greater than the logical symbol **0**.

Table of Content

**Boolean Algebra Formula**

Following are the operations of Boolean algebra:

**OR Operation**

The symbol ‘**+**’ denotes the **OR** operator.

To perform this operation we need a minimum of 2 input variables that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0/1).

**OR** operation is defined for **A OR B** or **A+B** as if **A = B = 0** then **A+B = 0** or else **A+B = 1.**

The result of an **OR** operation is equal to the input variable with the greatest value.

Following are the possible outputs with a minimum of 2 input combinations:

**0 + 0 = 0****0 + 1 =1****1 + 0 = 1****1 + 1 = 1**

**AND Operation**

The symbol ‘**.**’ denotes the **AND** operator.

To perform this operation we need a minimum of 2 input variables that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0/1).

**And** operation is defined for **A AND B** or **A.B**, if **A = B = 1** then **A.B = 1** or else **A.B = 0.**

The result of an **AND** operation is equal to the input variable with the lowest value.

Following are the possible outputs with a minimum of 2 input combinations:

**0 .0= 0****0 .1 = 0****1 .0 = 0****1 .1 = 1**

**Not Operation**

Not operation is also known as **Complement Operation**. This is a special operation which is denoted by ‘**’**’. Complement of A is represented as **A’**.

To perform this operation we need a minimum of 1 input variable that can take the values of binary numbers i.e., 0 or 1 to get an output with one binary value (0/1).

Not operation is defined for **A’** or **NOT A** if **A = 1**, then **A’ = 0** or else **A’ = 1**.

The result of a not operation is the inverse value of the set of inputs provided.

Following are the possible outputs with minimum 1 input value:

**(1)’ = 0****(0)’ = 1**

**Boolean Algebra Rules**

The Following are the important rules followed in Boolean algebra.

- Input variables used in Boolean algebra can take the values of binary numbers i.e.,
**0**or**1**. Binary number**1 is**for**HIGH**and Binary**0**is for**LOW**. - The complement/negation/inverse of a variable is represented by
**‘**

Thus, the complement of variable**A**is represented as**A’**. Thus**A’**if**A = 1**, then**A’ = 0**or else**A’ = 1** **OR**-ing of the variables is represented by a ‘**+**’ sign between them. For example,**OR**-ing of**A**,**B**is represented as**A + B**.- Logical
**AND**-ing of two or more variables is represented by a ‘**.**’ sign between them, such as**A.B**. Sometime the ‘**.**’ may be omitted like**AB**.

**Boolean Laws**

**Commutative law**

Binary operation(s) satisfying anyone of the following expressions is/are said to a commutative operation.

**A.B = B.A****A+B = B+A**

Commutative law states that** changing the sequence of the variables does not have any effect on the output of a logic circuit**.

**Associative law**

Law of Associative states that **the order in which the logic operations are performed is irrelevant as their effect is the same.**

**(A.B).C = A.(B.C)****(A+B)+C = A+(B+C)**

**Distributive law**

**Law of Distributive states the following conditions**

**A.(B+C) = A.B + A.C****A+(B.C) = (A+B).(A+C)**

**AND law**

**AND Law states the following conditions as they are using AND Operations.**

**A.0 = 0****A.1 = A****A.A = A****A.A’ = 0**

**OR law**

**OR laws states the following conditions as they are using OR Operations.**

**A+0 = A****A+A = A****A+1 = 1****A+A’ = A**

**Complement law**

This law uses the NOT operation. The law of Complement also known as Inversion/Negation, states that **double inversion of a variable result in the original variable itself.**

**(A’)’ = A**or**A + (A’)’ = 1**

**Boolean Algebra Truth Table**

#### Truth Table of AND Operation and OR Operation

A | B | A+B | A.B |
---|---|---|---|

0 | 0 | 0 | 0 |

0 | 1 | 1 | 0 |

1 | 0 | 1 | 0 |

1 | 1 | 1 | 1 |

**Truth Table of NOT operation**

A | A’ |
---|---|

0 | 1 |

1 | 0 |

**Truth Table of NOT operation**

**Logic Gates**

Digital systems are said to be built using Logic Gates. A Logic gate is an electronic circuit or logic circuit which can take one or more than one input to get only one output. A particular logic is the relationship between the inputs and the output of a logic gate.

**Types of Logic gates**

**AND Gate**

**This logic gate uses AND operation logic and denoted by**

Input (A) | Input (B) | Output |
---|---|---|

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

**Truth Table**of

**AND Gate**

**OR Gate**

**This logic gate uses OR operation logic and denoted by**

A | B | Output |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

**Truth Table**of

**OR Gate**

**NOT Gate**

**This logic gate uses NOT operation logic & denoted by**

**It is also known as an Inverter.**

Input | Output |
---|---|

0 | 1 |

1 | 0 |

**NAND Gate**

A NOT-AND operation is known as NAND operation, and a logic gate using this NAND operation logic is called NAND gate.

Here the output of AND gate is the input of the NOT gate and the output of this combination of **NOT gate** and **AND gate** is the output of the **NAND gate**.

**NAND Gate** Diagram

**NOR Gate**

A **NOT-OR** operation is known as **NOR** operation, and a logic gate using this **NOR** operation logic is called the **NOR gate**.

Here the output of the **OR** gate is the input of the **NOT** gate and the output of this combination of **NOT** **gate** and **OR** **gate** is the output of the **NOR gate**.

**NOR Gate Diagram**

Input (A) | Input (B) | Output ()A ⊕ B |
---|---|---|

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 0 |

**XOR Gate**

**XOR** or **EXOR** or **Exclusive-OR** is a special type of gate or circuit which will give high output if even or zero number of inputs are high or else it will give low output.

The algebraic expressions and both represent the XOR gate with inputs *A* and *B*.

The Operation of this gate is denoted by **⊕**

**XOR Gate** **Logic**

**D = A XOR B****D = A ⊕ B ****D = A’.B + A.B’**

**XOR** **Logic Diagram**

**XOR Truth Table**

Input (A) | Input (B) | Output ()A ⊕ B |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

**XNOR Gate**

**XNOR** or **EX-NOR** or **Exclusive NOR** gate is a special type of gate** **or circuit that will give high output if an odd number of inputs are high or else it will give low output. It is the opposite of the XOR gate.

The Operation of this gate is denoted by ‘**Ɵ’**.

**XNOR Gate** **Logic**

**D = A Ɵ** **B****D = A’.B’ + A.B**

**XNOR Gate Diagram**

**XNOR Truth **Table

Input (A) | Input (B) | Output (A Ɵ B) |
---|---|---|

0 | 0 | 1 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |